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We (a) prove that continuous morphisms from locally compact groups to locallyexponential (possibly infinite-dimensional) Lie groups factor through Liequotients, recovering a result of Shtern's on factoring norm-continuousrepresentations on Banach spaces; (b) characterize the maximalalmost-periodicity of the identity component $\mathbb{G}_0\le \mathbb{G}$ of alocally compact group in terms of sufficiently discriminating families ofcontinuous functions on $\mathbb{G}$ valued in Hausdorff spaces generalizing ananalogous result by Kadison-Singer; (c) apply that characterization to recoverthe von Neumann kernel of $\mathbb{G}_0$ as the joint kernel of allappropriately bounded and continuous $\mathbb{G}$-representations ontopological vector spaces extending Kallman's parallel statement for unitaryrepresentations, and (d) provide large classes of complete locally convextopological vector spaces (e.g. arbitrary products of Fr\'echet spaces) withthe property that compact-group representations thereon whose vectors all havefinite-dimensional orbits decompose as finite sums of isotypic components. Thislast result specializes to one of Hofmann-Morris on representations on productsof real lines.

Factorization and boundedness for representations of locally compact groups on topological vector spaces